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Theorem frgrwopreglem5lem 29573
Description: Lemma for frgrwopreglem5 29574. (Contributed by AV, 5-Feb-2022.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtxβ€˜πΊ)
frgrwopreg.d 𝐷 = (VtxDegβ€˜πΊ)
frgrwopreg.a 𝐴 = {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}
frgrwopreg.b 𝐡 = (𝑉 βˆ– 𝐴)
frgrwopreg.e 𝐸 = (Edgβ€˜πΊ)
Assertion
Ref Expression
frgrwopreglem5lem (((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ((π·β€˜π‘Ž) = (π·β€˜π‘₯) ∧ (π·β€˜π‘Ž) β‰  (π·β€˜π‘) ∧ (π·β€˜π‘₯) β‰  (π·β€˜π‘¦)))
Distinct variable groups:   π‘₯,𝑉   π‘₯,𝐴   π‘₯,𝐺   π‘₯,𝐾   π‘₯,𝐷   𝐴,𝑏   π‘₯,𝐡   𝑦,𝐷   𝐺,π‘Ž,𝑏,𝑦,π‘₯   𝑦,𝑉
Allowed substitution hints:   𝐴(𝑦,π‘Ž)   𝐡(𝑦,π‘Ž,𝑏)   𝐷(π‘Ž,𝑏)   𝐸(π‘₯,𝑦,π‘Ž,𝑏)   𝐾(𝑦,π‘Ž,𝑏)   𝑉(π‘Ž,𝑏)

Proof of Theorem frgrwopreglem5lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.a . . . . . 6 𝐴 = {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾}
21reqabi 3455 . . . . 5 (π‘₯ ∈ 𝐴 ↔ (π‘₯ ∈ 𝑉 ∧ (π·β€˜π‘₯) = 𝐾))
3 fveqeq2 6901 . . . . . . 7 (π‘₯ = π‘Ž β†’ ((π·β€˜π‘₯) = 𝐾 ↔ (π·β€˜π‘Ž) = 𝐾))
43, 1elrab2 3687 . . . . . 6 (π‘Ž ∈ 𝐴 ↔ (π‘Ž ∈ 𝑉 ∧ (π·β€˜π‘Ž) = 𝐾))
5 eqtr3 2759 . . . . . . . . 9 (((π·β€˜π‘Ž) = 𝐾 ∧ (π·β€˜π‘₯) = 𝐾) β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯))
65expcom 415 . . . . . . . 8 ((π·β€˜π‘₯) = 𝐾 β†’ ((π·β€˜π‘Ž) = 𝐾 β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯)))
76adantl 483 . . . . . . 7 ((π‘₯ ∈ 𝑉 ∧ (π·β€˜π‘₯) = 𝐾) β†’ ((π·β€˜π‘Ž) = 𝐾 β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯)))
87com12 32 . . . . . 6 ((π·β€˜π‘Ž) = 𝐾 β†’ ((π‘₯ ∈ 𝑉 ∧ (π·β€˜π‘₯) = 𝐾) β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯)))
94, 8simplbiim 506 . . . . 5 (π‘Ž ∈ 𝐴 β†’ ((π‘₯ ∈ 𝑉 ∧ (π·β€˜π‘₯) = 𝐾) β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯)))
102, 9biimtrid 241 . . . 4 (π‘Ž ∈ 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯)))
1110imp 408 . . 3 ((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯))
1211adantr 482 . 2 (((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π·β€˜π‘Ž) = (π·β€˜π‘₯))
13 frgrwopreg.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
14 frgrwopreg.d . . . 4 𝐷 = (VtxDegβ€˜πΊ)
15 frgrwopreg.b . . . 4 𝐡 = (𝑉 βˆ– 𝐴)
1613, 14, 1, 15frgrwopreglem3 29567 . . 3 ((π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡) β†’ (π·β€˜π‘Ž) β‰  (π·β€˜π‘))
1716ad2ant2r 746 . 2 (((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π·β€˜π‘Ž) β‰  (π·β€˜π‘))
18 fveqeq2 6901 . . . . . 6 (π‘₯ = 𝑧 β†’ ((π·β€˜π‘₯) = 𝐾 ↔ (π·β€˜π‘§) = 𝐾))
1918cbvrabv 3443 . . . . 5 {π‘₯ ∈ 𝑉 ∣ (π·β€˜π‘₯) = 𝐾} = {𝑧 ∈ 𝑉 ∣ (π·β€˜π‘§) = 𝐾}
201, 19eqtri 2761 . . . 4 𝐴 = {𝑧 ∈ 𝑉 ∣ (π·β€˜π‘§) = 𝐾}
2113, 14, 20, 15frgrwopreglem3 29567 . . 3 ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡) β†’ (π·β€˜π‘₯) β‰  (π·β€˜π‘¦))
2221ad2ant2l 745 . 2 (((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π·β€˜π‘₯) β‰  (π·β€˜π‘¦))
2312, 17, 223jca 1129 1 (((π‘Ž ∈ 𝐴 ∧ π‘₯ ∈ 𝐴) ∧ (𝑏 ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ ((π·β€˜π‘Ž) = (π·β€˜π‘₯) ∧ (π·β€˜π‘Ž) β‰  (π·β€˜π‘) ∧ (π·β€˜π‘₯) β‰  (π·β€˜π‘¦)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  {crab 3433   βˆ– cdif 3946  β€˜cfv 6544  Vtxcvtx 28256  Edgcedg 28307  VtxDegcvtxdg 28722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552
This theorem is referenced by:  frgrwopreglem5  29574
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